Suppose that .
If then by definition of supremum and infimum is not an upper bound of A, so but c is not a lower bound of B. That means , but that means d is a upper bound of A which is impossible because . That means that .
This time suppose that which implies .
But this means that is an upper bound for A but it cannot belong to B. That is a contradiction. Thus .

Suppose that .
If then by definition of supremum and infimum is not an upper bound of A, so but c is not a lower bound of B. That means , but that means d is a upper bound of A which is impossible because . That means that .
This time suppose that which implies .
But this means that is an upper bound for A but it cannot belong to B. That is a contradiction. Thus .